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Theorem sbieh 1670
Description: Conversion of implicit substitution to explicit substitution. New proofs should use sbie 1671 instead. (Contributed by NM, 30-Jun-1994.) (New usage is discouraged.)
Hypotheses
Ref Expression
sbieh.1 (ψxψ)
sbieh.2 (x = y → (φψ))
Assertion
Ref Expression
sbieh ([y / x]φψ)

Proof of Theorem sbieh
StepHypRef Expression
1 id 19 . 2 (φφ)
21hbth 1349 . . 3 ((φφ) → x(φφ))
3 sbieh.1 . . . 4 (ψxψ)
43a1i 9 . . 3 ((φφ) → (ψxψ))
5 sbieh.2 . . . 4 (x = y → (φψ))
65a1i 9 . . 3 ((φφ) → (x = y → (φψ)))
72, 4, 6sbiedh 1667 . 2 ((φφ) → ([y / x]φψ))
81, 7ax-mp 7 1 ([y / x]φψ)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1240  [wsb 1642
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-i9 1420  ax-ial 1424
This theorem depends on definitions:  df-bi 110  df-sb 1643
This theorem is referenced by:  sbie  1671  sbco2vlem  1817  equsb3lem  1821  sbco2yz  1834  elsb3  1849  elsb4  1850  dvelimf  1888
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